Download Probability Models in Computer Science

Snehashu Saha, Probabiity Models
1
Probability Models in
Computer Science
@Reva University
04/11/2015
Snehanshu Saha, PESIT South Campus
Concepts
2

The Idea of Probability

Probability Models

Probability Rules

Finite and Discrete Probability Models

Continuous Probability Models
Snehashu Saha, Probabiity Models
04/11/2015
The Idea of Probability
3
Chance behavior is unpredictable in the short run, but has a regular and
predictable pattern in the long run.
We call a phenomenon random if individual outcomes are uncertain but
there is nonetheless a regular distribution of outcomes in a large number
of repetitions.
The probability of any outcome of a chance process is the proportion of
times the outcome would occur in a very long series of repetitions.
04/11/2015BPS - 5th
Ed.
Snehashu Saha, Probabiity
ModelsChapter 5
3
Myths About Randomness
4
The idea of probability seems straightforward. However, there are
several myths of chance behavior we must address.
The myth of short-run regularity:
The idea of probability is that randomness is predictable in the long
run. Our intuition tries to tell us random phenomena should also be
predictable in the short run. However, probability does not allow us to
make short-run predictions.
The myth of the “law of averages”:
Probability tells us random behavior evens out in the long run. Future
outcomes are not affected by past behavior. That is, past outcomes
do not influence the likelihood of individual outcomes occurring in the
future.
Snehashu Saha, Probabiity Models
04/11/2015
Probability Models
5
Descriptions of chance behavior contain two parts: a list of possible
outcomes and a probability for each outcome.
The sample space S of a chance process is the set of all possible
outcomes.
An event is an outcome or a set of outcomes of a random
phenomenon. That is, an event is a subset of the sample space.
A probability model is a description of some chance process that
consists of two parts: a sample space S and a probability for each
outcome.
Snehashu Saha, Probabiity Models
04/11/2015
Probability Models
6
Example: Give a probability model for the chance process of rolling two fair, sixsided dice―one that’s red and one that’s green.
Sample Space
36 Outcomes
Since the dice are fair, each outcome is equally
likely.
Each outcome has probability 1/36.
Snehashu Saha, Probabiity Models
04/11/2015
Probability Rules
7
1. Any probability is a number between 0 and 1.
2. All possible outcomes together must have probability 1.
3. If two events have no outcomes in common, the probability that
one or the other occurs is the sum of their individual probabilities.
4. The probability that an event does not occur is 1 minus the
probability that the event does occur.
Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.
Rule 2. If S is the sample space in a probability model, then P(S) = 1.
Rule 3. If A and B are disjoint, P(A or B) = P(A) + P(B).
This is the addition rule for disjoint events.
Rule 4: For any event A, P(A does not occur) = 1 – P(A).
04/11/2015BPS - 5th
Ed.
Snehashu Saha, Probabiity
ModelsChapter 5
7
Probability Rules
8
Distance-learning courses are rapidly gaining popularity among college
students. Randomly select an undergraduate student who is taking
distance-learning courses for credit and record the student’s age. Here
is the probability model:
Age group (yr):
Probability:
18 to 23
24 to 29
30 to 39
40 or over
0.57
0.17
0.14
0.12
(a) Show that this is a legitimate probability model.
Each probability is between 0 and 1 and
0.57 + 0.17 + 0.14 + 0.12 = 1
(b) Find the probability that the chosen student is not in the
traditional college age group (18 to 23 years).
P(not 18 to 23 years) = 1 – P(18 to 23 years)
= 1 – 0.57 = 0.43
Snehashu Saha, Probabiity Models
04/11/2015
Finite and Discrete Probability
Models
9
One way to assign probabilities to events is to assign a probability to
every individual outcome, then add these probabilities to find the
probability of any event. This idea works well when there are only a
finite (fixed and limited) number of outcomes.
A probability model with a finite sample space is called finite.
To assign probabilities in a finite model, list the probabilities of
all the individual outcomes. These probabilities must be
numbers between 0 and 1 that add to exactly 1. The probability
of any event is the sum of the probabilities of the outcomes
making up the event.
Snehashu Saha, Probabiity Models
04/11/2015
Random Variables
10
A probability model describes the possible outcomes of a chance process and
the likelihood that those outcomes will occur.
A numerical variable that describes the outcomes of a chance process is called
a random variable. The probability model for a random variable is its
probability distribution.
A random variable takes numerical values that describe the outcomes
of some chance process.
The probability distribution of a random variable gives its possible
values and their probabilities.
Example: Consider tossing a fair coin 3 times.
Define X = the number of heads obtained
X = 0: TTT
X = 1: HTT THT TTH
X = 2: HHT HTH THH
X = 3: HHH
Value
0
1
2
3
Probability
1/8
3/8
3/8
1/8
Snehashu Saha, Probabiity Models
04/11/2015
Discrete Random Variable
11
There are two main types of random variables: discrete and
continuous. If we can find a way to list all possible outcomes for a
random variable and assign probabilities to each one, we have a
discrete random variable.
A discrete random variable X takes a fixed set of possible values with gaps
between. The probability distribution of a discrete random variable X lists the values
xi and their probabilities pi:
Value:
Probability:
x1
p1
x2
p2
x3
p3
…
…
The probabilities pi must satisfy two requirements:
1.Every probability pi is a number between 0 and 1.
2.The sum of the probabilities is 1.
To find the probability of any event, add the probabilities pi of the particular values xi
that make up the event.
Snehashu Saha, Probabiity Models
04/11/2015
Continuous Random Variable
12
Discrete random variables commonly arise from situations that involve
counting something. Situations that involve measuring something often
result in a continuous random variable.
A continuous random variable Y takes on all values in an interval of
numbers. The probability distribution of Y is described by a density
curve. The probability of any event is the area under the density curve
and above the values of Y that make up the event.
The probability model of a discrete random variable X assigns a
probability between 0 and 1 to each possible value of X.
A continuous random variable Y has infinitely many possible values.
All continuous probability models assign probability 0 to every
individual outcome. Only intervals of values have positive probability.
Snehashu Saha, Probabiity Models
04/11/2015
Allocation and utilization-fixed
resource
13

Advanced applications:
Snehashu Saha, Probabiity Models
04/11/2015
Probability modeling-load balancing
14






Delay estimation theorem
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
www.guruchandali.com
???????????
Quality of Service gurantees, Load estimation…
Lets move to Blackboard…
Snehashu Saha, Probabiity Models
04/11/2015
Queue length and waiting time
15
Snehashu Saha, Probabiity Models
04/11/2015
A Mobility-Probabilistic search engine
16

“Friends” in Property search engine
Snehashu Saha, Probabiity Models
04/11/2015